210 XIII INTEGRATION 3. (a) Let p be a positive measure on a locally compact space X, and let A be a /x-integ- rable set such that /x(A) > 0. Suppose that, for each /z-integrable subset B of A, we have either /x(B) = 0 or ju,(B) = ju,(A). Show that there exists a point a e A such that (Ji({a}) = ju(A). (Consider the intersection of the compact sets K <= A such that ^(K) = fi(A): show that it is not empty, has measure equal to ju,(A) and consists of a single point.) (b) Suppose that ju is a diffuse measure. For each p-integrable set A such that /x(A) > 0 and each e > 0, show that there exists a ju,-integrable subset B of A such that 0 < ju,(B) <i e (use (a) to show that there exists an integrable subset C of A such that 0 < ju,(C) ^ £ju(A)). Deduce that, as B runs through the set of /x-integrable subsets of A, the set of values of ju,(B) is the closed interval [0, /x(A)]. (For each real number b such that 0 < b < ft(A), let c be the least upper bound of the measures of measurable subsets C of A such that /z(C) <; b. Show that c = b by using the preceding result, and then show that there exists an increasing sequence (Cn) of measurable subsets of A such that lim ju-(Cn) = b.) n-»oo 4. (a) Let v be a positive atomic measure on a locally compact space X, and let A be a v-integrable subset of X. Show that the set of values of v(B), as B runs through the set of v-integrable subsets of A, is closed in R. (Let P be the smallest set carry- ing v. Assuming that A n P is infinite, and arranging the points of A n P in a sequence (*„), consider the mapping y of the product space {0, 1}N into R defined by 9(0= S £nv({xn})t and show that 99 is continuous.) n (b) Deduce from (a) and Problem 3(b) that if JJL is any positive measure on X and if A is a /Lc-integrable subset of X, then the set of values of ju,(B), as B runs through the set of /A-integrable subsets of A, is closed in R. Extend this result to the situation where /x is any real measure on X. (c) Deduce from Problem 3(b) that if /u, is a diffuse real measure on X, then the set of values of /u(A) = ^+(A) — /x~(A), where A runs through the set of | p, \ -integrable sub- sets of X, is a closed (possibly unbounded) interval of R. (d) Give an example of an atomic positive measure v on a locally compact, non- compac.t space X, such that the set of values of v(A), where A runs through the set of v-integrable subsets of X, is not closed in R. (Take v so that inf v({x}) > 0.) xeX 5. (a) Let X be a compact space and ju, =£ 0 a diffuse positive measure on X. Let (/„) be a total orthonormal sequence in «^c(X> ju). Show thatj] |/nt*)|2 = 4-00 for almost n all xeX. (Argue by contradiction, using Problem 3(b), Bessel's inequality, and the Cauchy-Schwarz inequality for series: show that there would exist a measurable set B with measure >0 and arbitrarily small, contained in the set of points x such that £ \fn(x)\z < +00, such that, if cn = ((pB !/„), the series £ cnfn(x) converges almost n n everywhere in B to a function < J-; then observe that by virtue of the hypothesis that the sequence (/„) is total, together with Parseval's identity, we have lim f |1 - sn(x)\2 4t(x) = 0, where sn = £ ckfk . «-*oo JB fc=l Hence obtain a contradiction.) (b) Under the same hypotheses, show that there exists a sequence (bn) of scalars such that lim bn = 0 and £ |6,,|2|/n(x)|2 = 4-00 almost everywhere. (Use EgorofFs theoremby translation (cf. (14.1)).ting of a single point is closed,